Summary and Analysis

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Computing Derivatives

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In this section we compute the derivatives of the elementary functions. We use the
definition of the derivative as a limit of difference quotients. Recall that a
function
f
is said to be differentiable at a value
x
in its domain if the limit

exists, and that the value of this limit is called the
derivative of
f
at
x
.

Derivatives of Linear Functions

A linear function has the form
f (x) = ax + b
. Since the slope of this line is
a
, we would expect the derivative
f'(x)
to equal
a
at every point in its domain. Computing the limit of the
difference quotient, we see that this is the case:

f'(x)

=

=

=

=

a

=

a

Thus the graph of the derivative is the horizontal line
f'(x) = a
.

Note, as a special case, that the derivative of any constant function
f (x) = b
is a constant
function equal to
0
at every value in its domain:
f'(x) = 0
.

Derivatives of Polynomial Functions

We will show in the next section
that the derivative of a sum of two functions is equal to the sum of the
derivatives of the two functions. For example, considering the linear function
f
above, let
f0(x) = b
and
f1(x) = ax
. Then
f (x) = f0(x) + f1(x)
, so
f'(x) = f0'(x) + f1'(x) = a + 0 = a
, agreeing with our previous result.

In studying polynomial functions, it is
therefore enough to find the derivative of a monomial function of the form
f (x) = axn
. Plugging into the formula for the derivative, we have

f'(x)

=

=

=

=

a[nxn-1 + xn-2Δx + ... + Δxn-1]

=

anxn-1

Thus, to take the derivative of a monomial function, we multiply by the exponent and
reduce the exponent by
1
. Using the property of the derivative mentioned above, we
see that the derivative of the polynomial function
f (x) = anxn + ... + a1x + a0
is
given by
f (x) = nanxn-1 + ... + a2x + a1
.

We will wait until we have the quotient rule at our disposal before we calculate the
derivatives of rational functions.

Derivatives of Power Functions

A power function has the form
f (t) = Crt
. Plugging into the formula for the derivative, we have

f'(t)

=

=

=

=

Crt

The limit in the final expression above does not depend on
t
, so it is a
constant. In fact, this limit is one way of defining the value of the natural
logarithm function at
r
, or
log(r)
. Thus we have

f'(t) = Crtlog(r)

In the special case where
r = e
, where
e
is the number such that
log(e) = 1
, we
have f'(t)=f(t). The functions
f (t) = Cet
are the only functions
that are equal to their own derivatives.

Derivatives of Trigonometric Functions

We now give one way of calculating the derivative of the sine function. Let
f (x) = sin(x)
.
Using the trigonometric identity
sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
, we have

f'(x)

=

=

=

=

sin(x) + cos(x)

=

cos(x)

where the last equality follows from examining the figure below:

Figure %: Calculating the Derivative of the Sine Function

We may similarly compute the derivative of
g(x) = cos(x)
to be
g'(x) = - sin(x)
.
Finally, since
tan(x) = sin(x)/cos(x)
, it will follow from the quotient rule that the
derivative of
h(x) = tan(x)
is
h'(x) = 1/(cos(x))2
.

We will compute the derivatives of the inverse trigonometric functions in the next
section, using implicit differentiation.